|
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integral coefficients. A lattice may be viewed as a regular tiling of a space by a primitive cell. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Calabi-Yau manifold Geometry (Greek γεωμετÏία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...
Group theory is that branch of mathematics concerned with the study of groups. ...
In mathematics, a discrete group is a group G equipped with the discrete topology. ...
In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...
In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...
The integers are commonly denoted by the above symbol. ...
Plane tilings by regular polygons have been widely used since antiquity. ...
In solid state physics and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum volume cell corresponding to a single lattice point. ...
Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory. They also arise in applied mathematics in connection with coding theory, and are used in various ways in the physical sciences. For instance, in materials science and solid-state physics, a lattice is a synonym for a crystalline structure, a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal. More generally, lattice models are studied in physics, often by the techniques of computational physics. In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...
Coding theory is a branch of mathematics and computer science dealing with the error-prone process of transmitting data across noisy channels, via clever means, so that a large number of errors that occur can be corrected. ...
The Materials Science Tetrahedron, which often also includes Characterization at the center Materials science is an interdisciplinary field involving the properties of matter and its applications to various areas of science and engineering. ...
Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ...
In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...
Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle still characterizing a chemical element. ...
In science, a molecule is a group of atoms in a definite arrangement held together by chemical bonds. ...
Quartz crystal Synthetic bismuth hopper crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid in which the constituent atoms...
In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. ...
The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ...
Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which a quantitative theory already exists. ...
Symmetry considerations and examples
A lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. The symmetry group of an object (e. ...
A translation slides an object by a vector a: Ta(p) = p + a. ...
A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translate of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense. 2-dimensional renderings (ie. ...
Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle still characterizing a chemical element. ...
In science, a molecule is a group of atoms in a definite arrangement held together by chemical bonds. ...
Quartz crystal Synthetic bismuth hopper crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid in which the constituent atoms...
In mathematics, a symmetry group describes all symmetries of objects. ...
In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...
A simple example of a lattice in Rn is the subgroup Zn. A more complicated example is the Leech lattice, which is a lattice in R24. The period lattice in R2 is central to the study of elliptic functions, developed in nineteenth century mathematics; it generalises to higher dimensions in the theory of abelian functions. In mathematics, the Leech lattice is a lattice Λ in R24 discovered John Leech ( 16 (1964), 657--682). ...
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ...
In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ...
Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...
For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ...
Dividing space according to a lattice A typical lattice Λ in Rn thus has the form  where {v1, ..., vn} is a basis for Rn. Different bases can generate the same lattice, but the absolute value of the determinant of the vectors vi is uniquely determined by Λ, and is denoted by d(Λ). If one thinks of a lattice as dividing the whole of Rn into equal polyhedra (copies of an n-dimensional parallelepiped, known as the fundamental region of the lattice), then d(Λ) is equal to the n-dimensional volume of this polyhedron. This is why d(Λ) is sometimes called the covolume of the lattice. In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ...
In geometry, a parallelepiped (now usually pronounced , traditionally[1] in accordance with its etymology in Greek παÏαλληλ-επίπεδον, a body having parallel planes) is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...
In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ...
The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ...
Lattice points in convex sets Minkowski's theorem relates the number d(Λ) and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d(Λ) as well. In mathematics, Minkowskis theorem in the geometry of numbers applies to convex symmetric sets and lattices; it relates the number of contained lattice points to the volume of such a set. ...
Look up Convex set in Wiktionary, the free dictionary. ...
In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...
In mathematics, integral polytopes have associated Ehrhart polynomials which encode some geometrical information about them. ...
Computing with lattices Lattice basis reduction is the problem of finding a short lattice basis. The Lenstra-Lenstra-Lovász lattice reduction algorithm (LLL) finds a short lattice basis in polynomial time; it has found numerous applications, particularly in public-key cryptography. The Lenstra-Lenstra-Lovász lattice reduction (LLL) algorithm is an algorithm which, given a lattice basis as input, outputs a basis with short, nearly orthogonal vectors. ...
In computational complexity theory, polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ...
A big random number is used to make a public-key pair. ...
Lattices in two dimensions: detailed discussion There are five 2D lattice types as given by the crystallographic restriction theorem. Below the wallpaper group of the lattice is given in parentheses; note that a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. If the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. ...
Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. ...
- a rhombic lattice, also called centered rectangular lattice or isosceles triangular lattice (cmm), with evenly spaced rows of evenly spaced points, with the rows alternatingly shifted one half spacing (symmetrically staggered rows); special cases are:
- a hexagonal lattice or equilateral triangular lattice (p6m)
- a square lattice (see below, and turn 45°)
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * - more generally, a parallelogrammic lattice, also called oblique lattice (p2)(with asymmetrically staggered rows):
* * * * * * * * * * * * * * * * * * * * * * * * * * * * For the classification of a given lattice, start with one point and take a nearest second point. For the third point, not on the same line, consider its distances to both points. Among the points for which the smaller of these two distances is least, choose a point for which the larger of the two is least. (Not logically equivalent but in the case of lattices giving the same result is just "Choose a point for which the larger of the two is least".) For other uses of the word rhombus, see Rhombus (disambiguation) This shape is a rhombus In geometry, a rhombus (or rhomb; plural rhombi) is a quadrilateral in which all of the sides are of equal length, i. ...
A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ...
Triangular tiling. ...
In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ...
Upright square tiling. ...
This article or section is not written in the formal tone expected of an encyclopedia article. ...
The five cases correspond to the triangle being equilateral, right isosceles, right, isosceles, and scalene. In a rhombic lattice, the shortest distance may either be a diagonal or a side of the rhombus, i.e., the line segment connecting the first two points may or may not be one of the equal sides of the isosceles triangle. This depends on the smaller angle of the rhombus being less than 60° or between 60° and 90°. A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ...
The general case is known as a period lattice. If the vectors p and q generate the lattice, instead of p and q we can also take p and p-q, etc. In general in 2D, we can take a p + b q and c p + d q for integers a,b, c and d such that ad-bc is 1 or -1. This ensures that p and q themselves are integer linear combinations of the other two vectors. Each pair p, q defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is a fundamental parallelogram. In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ...
For the crossed product in algebra and functional analysis, see crossed product. ...
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ...
The vectors p and q can be represented by complex numbers. Up to size and orientation, a pair can be represented by their quotient. Expressed geometrically: if two lattice points are 0 and 1, we consider the position of a third lattice point. Equivalence in the sense of generating the same lattice is represented by the modular group:  represents choosing a different third point in the same grid,  represents choosing a different side of the triangle as reference side 0-1, which in general implies changing the scaling of the lattice, and rotating it. Each "curved triangle" in the image contains for each 2D lattice shape one complex number, the grey area is a canonical representation, corresponding to the classification above, with 0 and 1 two lattice points that are closest to each other; duplication is avoided by including only half of the boundary. The rhombic lattices are represented by the points on its boundary, with the hexagonal lattice as vertex, and i for the square lattice. The rectangular lattices are at the imaginary axis, and the remaining area represents the parallelogrammetic lattices, with the mirror image of a parallelogram represented by the mirror image in the imaginary axis. Image File history File links Not the best picture, but its the only one I have right now. ...
Image File history File links Not the best picture, but its the only one I have right now. ...
In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ...
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ...
In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...
Lattices in three dimensions The 14 lattice types in 3D are called Bravais lattices. They are characterized by their space group. 3D patterns with translational symmetry of a particular type cannot have more, but may have less symmetry than the lattice itself. In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation operations. ...
The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ...
Lattices in complex space A lattice in Cn is a discrete subgroup of Cn which spans the 2n-dimensional real vector space Cn. For example, the Gaussian integers form a lattice in C. A Gaussian integer is a complex number whose real and imaginary part are both integers. ...
Every lattice in Rn is a free abelian group of rank n; every lattice in Cn is a free abelian group of rank 2n. In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ...
In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to contain it; or alternatively how large a free abelian group it can contain as a subgroup. ...
In Lie groups More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that the quotient G/Γ is of finite measure, for the measure on it inherited from Haar measure on G (left-invariant, or right-invariant - the definition is independent of that choice). That will certainly be the case when G/Γ is compact, but that sufficient condition is not necessary, as is shown by the case of the modular group in SL2(R), which is a lattice but where the quotient isn't compact (it has cusps). There are general results stating the existence of lattices in Lie groups. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...
In mathematics, a discrete group is a group G equipped with the discrete topology. ...
In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...
In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ...
Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...
In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...
A lattice is said to be uniform or cocompact if G/Γ is compact; otherwise the lattice is called non-uniform.
Lattices over general vector-spaces Whilst we normally consider  lattices in  this concept can be generalised to any finite dimensional vector space over any field. This can be done as follows: In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
Let K be a field, let V be an n-dimensional K-vector space, let  be a K-basis for V and let R be a ring contained within K. Then the R lattice  in V generated by B is given by: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In mathematics, a basis or set of generators is a collection of objects that can be systematically combined to produce a larger collection of objects. ...
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
 Different bases B will in general generate different lattices. However, if the transition matrix T between the bases is in GLn(R) - the general linear group of R (in simple terms this means that all the entries of T are in R and all the entries of T − 1 are in R - which is equivalent to saying that the determinant of T is in R * - the unit group of elements in R with multiplicative inverses) then the lattices generated by these bases will be isomorphic since T induces an isomorphism between the two lattices. In mathematics, a (discrete-time) Markov chain is a discrete-time stochastic process with the Markov property. ...
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ...
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In mathematics, a unit in a (unital) ring R is an invertible element of R, i. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
Important cases occur in number theory with K a p-adic field and R the p-adic integers. The title given to this article is incorrect due to technical limitations. ...
The title given to this article is incorrect due to technical limitations. ...
See also |
Feeds |
Contact